Nonnegative Tensor Factorization, Completely Positive Tensors and an Hierarchical Elimination Algorithm

Nonnegative tensor factorization has applications in statistics, computer vision, exploratory multiway data analysis and blind source separation. A symmetric nonnegative tensor, which has a symmetric nonnegative factorization, is called a completely positive (CP) tensor. The H-eigenvalues of a CP tensor are always nonnegative. When the order is even, the Z-eigenvalue of a CP tensor are all nonnegative. When the order is odd, a Z-eigenvector associated with a positive (negative) Z-eigenvalue of a CP tensor is always nonnegative (nonpositive). The entries of a CP tensor obey some dominance properties. The CP tensor cone and the copositive tensor cone of the same order are dual to each other. We introduce strongly symmetric tensors and show that a symmetric tensor has a symmetric binary decomposition if and only if it is strongly symmetric. Then we show that a strongly symmetric, hierarchically dominated nonnegative tensor is a CP tensor, and present a hierarchical elimination algorithm for checking this. Numerical examples are also given

Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. Furthermore, we prove that the Hadamard product of two positive semi-definite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised

Numerical Optimization for Symmetric Tensor Decomposition

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on real-valued decompositions, both unconstrained and nonnegative, for problems with low-rank structure. We discuss when solutions exist and how to formulate the mathematical program. Numerical results show the properties of the proposed formulations (including one that ignores symmetry) on a set of test problems and illustrate that these straightforward formulations can be effective even though the problem is nonconvex

Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors

Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we further introduce skew centrosymmetric tensors and centrosymmetric Cauchy tensors, and discuss properties of these three classes of structured tensors. Some sufficient and necessary conditions for a tensor to be centrosymmetric or skew centrosymmetric are given. We show that, a general tensor can always be expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric tensor. Some sufficient and necessary conditions for a Cauchy tensor to be centrosymmetric or skew centrosymmetric are also given. Spectral properties on H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and centrosymmetric Cauchy tensors are discussed. Some further questions on these tensors are raised